# K/L Decomposition: Numerator vs Denominator

**Reviewer Issue 3.3**: Investment/GDP rises at h=2 but Δlog(K/L) falls.
We decompose K/L into capital stock growth and employment growth.

Verification: corr(Δlog K/L, Δlog rnna − Δlog emp) = 1.000000

## Summary: Coefficient on log_predicted_demo_inflows by horizon

| Outcome | h=-3 | h=-2 | h=-1 | h=0 | h=1 | h=2 | h=3 | h=4 | h=5 |
|---------|------|------|------|-----|-----|-----|-----|-----|-----|
| Δlog(K/L) [capital per worker] | -0.0019 | -0.0006 | -0.0010 | 0.0064*** | -0.0060* | -0.0157*** | -0.0138*** | -0.0078* | -0.0082 |
| Δlog(rnna) [capital stock] | -0.0059*** | -0.0044*** | -0.0035*** | -0.0061*** | -0.0041*** | -0.0009 | 0.0001 | 0.0000 | -0.0032 |
| Δlog(emp) [employment] | -0.0026 | -0.0023 | -0.0015 | -0.0066*** | 0.0028 | 0.0148*** | 0.0152*** | 0.0103** | 0.0075 |
| Δlog(Y/L) [output per worker] | -0.0350*** | -0.0138** | -0.0135*** | -0.0119*** | 0.0011 | -0.0084 | -0.0133 | -0.0352*** | -0.0282*** |
| K/Y ratio [level] | -0.0364 | -0.1084** | -0.0670 | 0.1470*** | -0.2368*** | -0.1149** | -0.1445*** | 0.0644 | 0.0261 |
| Investment/GDP [baseline] | -0.2729 | -0.5237* | -0.8874*** | -1.6546*** | 0.1071 | 0.6917** | 0.2045 | -0.3329 | -0.5454 |

## Interpretation

At h=2:
- Capital stock growth (Δlog rnna): β=-0.0009, p=0.6094
- Employment growth (Δlog emp): β=0.0148, p=0.0004
- Capital per worker (Δlog K/L): β=-0.0157, p=0.0001
- Investment/GDP: β=0.6917, p=0.0437
- Output per worker (Δlog Y/L): β=-0.0084, p=0.2884

**Resolution**: Employment grows faster than capital stock, causing K/L to fall
despite rising investment. Demographic capital inflows attract both financial
and human capital (labor migration accompanies capital flows).


## ASCII Impulse Response Plots

```
Δlog(K/L) [capital per worker] (OECD)
=====================================

  h          coef       SE      p      N  Plot
  -          ----       --    ---    ---  ------------------------------------------------------------
 -3      -0.00195  0.00397 0.6235    703                         --------------·-------------         
 -2      -0.00058  0.00336 0.8631    703                              -----------·------------        
 -1      -0.00101  0.00203 0.6189    703                                  ------·-------              
  0    0.00636***  0.00214 0.0030    703                                          |   -------*--------
  1     -0.00596*  0.00337 0.0769    666                    ------------o-----------                  
  2   -0.01566***  0.00397 0.0001    629  -------------*--------------            |                   
  3   -0.01382***  0.00434 0.0015    592   ---------------*---------------        |                   
  4     -0.00775*  0.00465 0.0959    555             ----------------o----------------                
  5      -0.00817  0.00497 0.1011    518           -----------------·-----------------                
                                          ------------------------------------------------------------
        -0.0234                                            0.0105
  (* p<0.05, o p<0.1, · p>0.1, | = zero)

Δlog(rnna) [capital stock] (OECD)
=================================

  h          coef       SE      p      N  Plot
  -          ----       --    ---    ---  ------------------------------------------------------------
 -3   -0.00587***  0.00196 0.0029    703  ---------------*----------------       |                    
 -2   -0.00437***  0.00159 0.0061    703          -------------*-------------    |                    
 -1   -0.00352***  0.00098 0.0003    703                   -------*--------      |                    
  0   -0.00611***  0.00094 0.0000    703         -------*-------                 |                    
  1   -0.00412***  0.00145 0.0045    666             -----------*-----------     |                    
  2      -0.00094  0.00184 0.6094    629                      ---------------·--------------          
  3       0.00006  0.00219 0.9783    592                        -----------------·-----------------   
  4       0.00003  0.00250 0.9898    555                     --------------------·--------------------
  5      -0.00317  0.00262 0.2253    518       ---------------------·---------------------            
                                          ------------------------------------------------------------
        -0.0097                                            0.0049
  (* p<0.05, o p<0.1, · p>0.1, | = zero)

Δlog(emp) [employment] (OECD)
=============================

  h          coef       SE      p      N  Plot
  -          ----       --    ---    ---  ------------------------------------------------------------
 -3      -0.00260  0.00431 0.5462    703  --------------·--------------                               
 -2      -0.00228  0.00347 0.5117    703     -----------·-----------                                  
 -1      -0.00153  0.00187 0.4137    703           ------·------                                      
  0   -0.00665***  0.00187 0.0004    703   ------*------    |                                         
  1       0.00280  0.00327 0.3921    666              -----------·----------                          
  2    0.01478***  0.00413 0.0004    629                    |          --------------*-------------   
  3    0.01520***  0.00466 0.0012    592                    |         ---------------*----------------
  4     0.01031**  0.00494 0.0375    555                    |----------------*----------------        
  5       0.00753  0.00527 0.1535    518               -----------------·------------------           
                                          ------------------------------------------------------------
        -0.0110                                            0.0243
  (* p<0.05, o p<0.1, · p>0.1, | = zero)

Δlog(Y/L) [output per worker] (OECD)
====================================

  h          coef       SE      p      N  Plot
  -          ----       --    ---    ---  ------------------------------------------------------------
 -3   -0.03497***  0.00750 0.0000    703    -------------*-------------                 |             
 -2    -0.01377**  0.00619 0.0265    703                         -----------*-----------|             
 -1   -0.01347***  0.00313 0.0000    703                               -----*------     |             
  0   -0.01186***  0.00313 0.0002    703                                ------*-----    |             
  1       0.00112  0.00645 0.8621    666                                      -----------·------------
  2      -0.00838  0.00788 0.2884    629                           --------------·--------------      
  3      -0.01329  0.00910 0.1447    592                     ----------------·---------------         
  4   -0.03516***  0.00899 0.0001    555  ---------------*----------------              |             
  5   -0.02823***  0.00926 0.0024    518       ----------------*----------------        |             
                                          ------------------------------------------------------------
        -0.0528                                            0.0138
  (* p<0.05, o p<0.1, · p>0.1, | = zero)

K/Y ratio [level] (OECD)
========================

  h          coef       SE      p      N  Plot
  -          ----       --    ---    ---  ------------------------------------------------------------
 -3      -0.03644  0.04490 0.4173    703                        ----------·----------                 
 -2    -0.10840**  0.04519 0.0167    703               ----------*----------  |                       
 -1      -0.06698  0.04313 0.1209    703                     ---------·----------                     
  0    0.14703***  0.02789 0.0000    703                                      |         ------*-------
  1   -0.23675***  0.04268 0.0000    666  ---------*---------                 |                       
  2    -0.11490**  0.04479 0.0105    629               ----------*----------  |                       
  3   -0.14454***  0.04622 0.0019    592           ----------*-----------     |                       
  4       0.06444  0.04293 0.1340    555                                   ----------·---------       
  5       0.02612  0.04377 0.5509    518                               ----------·---------           
                                          ------------------------------------------------------------
        -0.3204                                            0.2017
  (* p<0.05, o p<0.1, · p>0.1, | = zero)

Investment/GDP [baseline] (OECD)
================================

  h          coef       SE      p      N  Plot
  -          ----       --    ---    ---  ------------------------------------------------------------
 -3      -0.27291  0.28318 0.3355    703                         ---------·---------                  
 -2     -0.52374*  0.27154 0.0542    703                     ---------o--------                       
 -1   -0.88743***  0.27252 0.0012    703               ---------*--------     |                       
  0   -1.65463***  0.30187 0.0000    703  ---------*----------                |                       
  1       0.10712  0.31194 0.7314    703                              ----------·----------           
  2     0.69170**  0.34238 0.0437    703                                      |-----------*-----------
  3       0.20450  0.34535 0.5539    703                              ------------·-----------        
  4      -0.33285  0.34401 0.3336    703                      -----------·-----------                 
  5      -0.54542  0.34358 0.1129    703                  -----------·-----------                     
                                          ------------------------------------------------------------
        -2.2463                                            1.3628
  (* p<0.05, o p<0.1, · p>0.1, | = zero)

```

*PanelGLS with AR(1). Controls: fiscal_bal_gdp, nfa_gdp_lag, log_rel_opw, kaopen.*
